Monotonicity, Convexity and Inequalities for the Generalized Complete (P, Q) - Elliptic Integrals

2018 ◽  
Vol 11 (2) ◽  
pp. 41-51
Author(s):  
Xia Song ◽  
Li Yin
Keyword(s):  
Elliptic Integrals

scholarly journals Master integrals for bipartite cuts of three-loop photon self energy

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
R. N. Lee ◽  
A. I. Onishchenko
Keyword(s):  
Spectral Density ◽  
High Energy ◽  
Elliptic Integrals ◽  
Iterated Integrals ◽  
Self Energy ◽  
Complete Elliptic Integrals ◽  
The One

Abstract We calculate the master integrals for bipartite cuts of the three-loop propagator QED diagrams. These master integrals determine the spectral density of the photon self energy. Our results are expressed in terms of the iterated integrals, which, apart from the 4m cut (the cut of 4 massive lines), reduce to Goncharov’s polylogarithms. The master integrals for 4m cut have been calculated in our previous paper in terms of the one-fold integrals of harmonic polylogarithms and complete elliptic integrals. We provide the threshold and high-energy asymptotics of the master integrals found, including those for 4m cut.

Some Applications of Elliptic Integrals of First Kind

2013 ◽  
Vol 65 (1) ◽  
Author(s):  
Yasamin Barakat ◽  
Nor Haniza Sarmin
Keyword(s):  
Algebraic Function ◽  
High Accuracy ◽  
Elliptic Integrals ◽  
Definite Integrals ◽  
Engineering Mathematics ◽  
Optimal Values ◽  
Complete Elliptic Integrals

One of the most important applications of elliptic integrals in engineering mathematics is their usage to solve integrals of the form  (Eq. 1), where  is a rational algebraic function and  is a polynomial of degree  with no repeated roots. Nowadays, incomplete and complete elliptic integrals of first kind are estimated with high accuracy using advanced calculators.  In this paper, several techniques are discussed to show how definite integrals of the form (Eq. 1) can be converted to elliptic integrals of the first kind, and hence be estimated for optimal values. Indeed, related examples are provided in each step to help clarification.

Remarks on generalized elliptic integrals

2009 ◽  
Vol 139 (2) ◽  
pp. 417-426 ◽  
Author(s):  
Xiaohui Zhang ◽  
Gendi Wang ◽  
Yuming Chu
Keyword(s):  
Elliptic Integrals ◽  
Complete Elliptic Integrals

We study the monotonicity for certain combinations of generalized elliptic integrals, thus generalizing analogous well-known results for classical complete elliptic integrals, and prove a conjecture put forward by Heikkala, Vamanamurthy and Vuorinen.

scholarly journals Quantum walks and elliptic integrals

2010 ◽  
Vol 20 (6) ◽  
pp. 1091-1098 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Random Walk ◽  
Generating Function ◽  
Elliptic Integral ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Elliptic Integrals ◽  
Two Dimensional ◽  
One Dimensional ◽  
Similar Expression ◽  
Return Probability

Pólya showed in his 1921 paper that the generating function of the return probability for a two-dimensional random walk can be written in terms of an elliptic integral. In this paper we present a similar expression for a one-dimensional quantum walk.

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